Monomials, polynomials and operations with them

We will give definitions of some of the basic concepts, which we will explain with concrete examples.

Definition 1: A rational expression that has no variables in the denominator is called an integer rational expression.

Definition 2: Monomial, we will call an integer rational expression, which is a product of letters and numbers. One's are also any number, variable or parameter.

Definition 3: We will say that a monomial is in normal form when it is written with only one numerical multiplier, which stands in first place and is called a quotient, and any product of ones is written as a power.

Example: Let's consider the monomial $3x.x.y.4.y.z$. Obviously, this monomial is not in normal form because in its notation we have two numerical factors $3$ and $4$, and also, products of equal letters that are not written as a power. The normal form of the monomial would be $3x.x.y.4.y.z=3.4.x.x.y.y.z=12x^2y^2z.$ As you can see, $12x^2y^2z$ is obviously in normal form because it satisfies all the conditions of Definition 3 - it has only one numerical multiplier $12$, and the products of equal letters is written as a power. 

Let us mention that if we have literal factors in a monomial denoting parameters, they are of the quotient of the monomial.

Definition 4: The degree of a monomial is called the sum of the degree indices of the variables in it.

Example: Consider the monomial $21x^3y^2zt.$ To calculate the degree of this monomial, we need to calculate the sum of the degrees of the variables $x$, $y$, $z$, and $t$, so the degree of the monomial is $3+2+1+1=7.$

Definition 5: Monomials that have the same normal form or differ only in their coefficients are called similar.

Example: In this case, we see that the two monomials do not have the same normal form because $x^2y^3\neq x^2y^2$ and therefore are not similar. However, if we consider the pair of monomials $3x^2y^3$ and $-10x^2y^3$ we see that they are similar because they differ only in their coefficients.

We can add and subtract monomials only if they are similar. Then we do the operations (add or subtract) with their coefficients, and just add the letters to the result.

Example: we have the monomials $5x^2y$ and $2x^2y$, so $5x^2y+2x^2y=(5+2)x^2y=7x^2y$. Let us now subtract $2x^2y$ from $5x^2y$, hence $5x^2y-2x^2y=(5-2)x^2y=3x^2y$.

Now let's see how we multiply and divide the ones.

Example: We multiply them and get $(10x^3y^2z^4).(-3x^2y^5z^2)=10.(-3).x^3.x^2.y^2.y^5.z^4.z^2=-30x^5y^7z^6$.

We may mention that when we multiply two or more ones, their product is again a one.

Example: We divide $24x^4y^3z$ by $6x^2y^2$, therefore $\frac{24x^4y^3z}{6x^2y^2}=4x^2yz.$ Here of course the variables $x$ and $y$ must be different from $0$.

Let us note that when dividing two ones, the resulting quotient is not always a one. For example $\frac{5x^2y^3}{3xyz}=\frac{5xy^2}{z}$.

Example: let's perform the grading $(2x^2y^3)^4.$ From the product grading property ($(a.b)^n=a^nb^n)$ we get that $(2x^2y^3)^4=2^4(x^2)^4(y^3)^4. $ Now we apply the power-ranging property ($(a^n)^m=a^{n.m}$) and get $2^4(x^2)^4(y^3)^4=16x^8y^{12}$.

Definition 6: An integer rational expression that is a sum of ones is called a polynomial.

Example: The expressions $3x^2-5x+6$, $xy+3$, $xyz+x^2y^3z+11xy-4$, etc. are examples of polynomials.

Definition 7: We say that a polynomial is made normal when it is represented as a collection of dissimilar monomials, each of which is in normal form. 

Example: The polynomial $x^4+3x^3-x^2+x-1$ is in normal form.

Definition 8: We will call the coefficients of a polynomial the coefficients of the monomials involved.

Definition 9: The highest of the powers of the monomials involved in the normal form of a polynomial is called the power of the polynomial.

Example: The degree of the polynomial $x^4+3x^3-x^2+x-1$ is $4$, because the highest degree of the singleton involved in the normal form of the given polynomial ($x^4$) is $4$.

Addition and subtraction of polynomials will be illustrated with the following example:

Example: Given the polynomials $A=3y^3+2y-3$ and $y^3-y-1$, therefore $A+B=3y^3+2y-3+y^3-y-1=(3y^3+y^3)+(2y-y)+(-3-1)=4y^3+y-4.$ Now let's find $A-B$. Subtract the two polynomials and we get $A-B=3y^3+2y-3-(y^3-y-1)=3y^3+2y-3-y^3+y+1=(3y^3-y^3)+(2y+y)+(-3+1)=2y^3+3y-2.$

We will illustrate the multiplication of a polynomial by a singular with the following example:

Example: $3x^3.(x^2-4x+2)=3x^3.x^2+3x^3.(-4x)+3x^2.2=3x^5-12x^4+6x^2.$

Finally, we will also give an example of multiplying a polynomial by a polynomial:

Example: $(x+3)(x^2-4x+2)=x.x^2+x.(-4x)+x.2+3.x^2+3.(-4x)+3.2=x^3-4x^2+2x+3x^2-12x+6.$ Now we bring the polynomial into normal form and get $x^3-x^2-10x+6.$

1 Problem Find the normal form of the polynomial $(y^3+3x)(x^3+3y)-x^3y^3.$

Solution: Unscramble the parentheses and perform the operations with the like monomials, hence $(y^3+3x)(x^3+3y)-x^3y^3=x^3y^3+3y^4+3x^4+9xy-x^3y^3=3x^4+3y^4+9xy.$

Problem 2 Calculate the value of the expression $4x^2(3x+8)-2x(6x^2+16x)$ at $x=1.01.$

Solution: Simplify the given expression literally and then substitute with $x=1.01$, so we get that $4x^2(3x+8)-2x(6x^2+16x)=12x^3+32x^2-12x^3-32x^2=0.$ Therefore, the value of the given expression is $0$ for each $x$ or in other words, the value of the expression does not depend on the values of $x.$

3 Problem Put the polynomial $A=4x-4a-x^3-3xa^2+a^3+3x^2a$ into normal form

Solution: Bring the polynomial into normal form, noting that $a$ is a parameter and not a variable, therefore order the monomials by powers of $x$ and obtain $A=-x^3+3ax^2+4x-3a^2x+a^3-4a=-x^3+3ax^2+(4-3a^2)x+(a^2-4)a.$ 

4 Problem Given the polynomial $A=3x^3-ax^2+ax+5x-2a+1.$

a) Bring the polynomial $A$ into normal form.

b) For which value of $a$ does the polynomial $A$ not have a term of degree one?

Solution: a) Bring the polynomial to normal form, noting that $a$ is a parameter, not a variable, hence we arrange the monomials by powers of $x$ and get $A=3x^3-ax^2+(a+5)x-2a+1$.

b) In order not to have a monodimension of degree one, the factor $(a+5)$ in front of $x$ must be equal to $0$, i.e. $a+5=0$, from where we get that $a=-5$.

5 Problem Find the sum of the polynomials $u=6xy^2+5x+7y-3$ and $v=5y^2x-3x-5y+3$.

Solution: Търсим $u+v=6xy^2+5x+7y-3+5xy^2-3x-5y+3=11xy^2+2x+2y$.

6 Problem Given the expression: 

$A=(x-y)(x^3+xy-y)-(x^2-1)(x-y^2+2)+x(y+y^2)$.

a) Reduce $A$ to its normal form.

b) Find the value of $A$ if $x=\frac{(-2)^3.(-27)^6}{9^9.8}$ and $\frac{3}{4}:\frac{9}{8}=y:\frac{1}{2}$.

Solution: a) Thus we obtain:

$A=x^3+x^2y-xy-x^2y-xy^2+y^2-$$-(x^3-x^2y^2+2x^2-x+y^2-2)+xy+xy^2$. We perform redundancy where possible and reveal the parentheses:

$A=x^3+y^2-x^3+x^2y^2-2x^2+x-y^2+2=x^2y^2-2x^2+x+2$.

b) Let us first calculate the values of $x$ and $y$. Имаме, че $x=\frac{(-2)^3.(-27)^6}{9^9.8}=-\frac{2^3.(3^3)^6}{(3^2)^9.2^3}=-\frac{3^{18}}{3^{18}}=-1$. For $y$ we have that $\frac{3}{4}:\frac{9}{8}=y:\frac{1}{2}$ hence $\frac{3}{4}.\frac{8}{9}=2y\implies 2y=\frac{2}{3}$ and $y=\frac{1}{2}$. Now we compute the value of $A$ for $x$ and $y$ found in this way:

$A=(-1)^2.(\frac{1}{3})^2-2.(-1)^2+(-1)+2=-\frac{8}{9}$.

7 Problem Prove that the value of the expression $A=b(b-2x)+x(x+b)-bx$ at $b=x+3$ does not depend on the values of $x$.

Solution:

$A=b^2-2bx+x^2+bx-bx$ $$implies$ $A=x^2-2bx+b^2$. Now substitute in the last equality $b=x+3$ and we get that: 

$A=x^2-2x(x+3)+(x+3)^2$

$A=x^2-2x^2-6x+(x+3)(x+3)$

$A=-x^2-6x+x^2+3x+3x+9$ and $A=9$.

Therefore, whatever value $x$ takes the given expression will always be equal to the constant $9$, i.e. the expression does not depend on the values of the variable.

8 Problem Given the expressions $M=(x-a)^2-ax(x+1)$ and $N=a(x^2+a)$. For which value of the parameter $a$ is the sum of the coefficients of the polynomial equal to $M-N$ $-9$?

Solution:

$M-N=(x-a)^2-ax(x+1)-a(x^2+a)$

$M-N=(x-a)(x-a)-ax^2-ax-ax^2-a^2$

$M-N=x^2-ax-ax+a^2-2ax^2-ax-a^2$

$M-N=1.x^2-2ax^2-3ax$.

Now after taking $x^2$ out before the parentheses of the first two monomials we get:

$M-N=(1-2a)x^2-3ax$.

The coefficients of the polynomial $M-N$ are in front of $x^2$ $(1-2a)$ and in front of $x$ $(-3a)$, respectively. From the condition, we want their sum $(1-2a)+(-3a)$ to be equal to $-9$. Thus we obtain the linear equation concerning the parameter $a$:

$1-2a-3a=-9$ $-5a=-10$ $a=2$. 

9 Problem For which value of the parameter $b$ in the normal form of the polynomial equal to $(x^2-3bx+b)(x^2-2x+3)$ does $x^2$ not contain?

Solution:

$A=x^4-2x^3+3x^2-3bx^3+6bx^2-9bx+bx^2-2bx+3b$

$A=x^4-2x^3-3bx^3+3x^2+7bx^2-11bx+3b$

$A=x^4+(-2-3b)x^3+(3+7b)x^2-11bx+3b$.

Now in order for $A$ not to contain a monomial with $x^2$, it is unnecessary for the coefficient in front of the monomial $(3+7b)x^2$ to be zero. Thus we arrive at the linear equation concerning the parameter $b$, which is $3+7b=0$ and from here we find that $b=-^frac{3}{7}$.

Homework assignments

1. Multiply the ones:

a) $A=-4x^2y^3$ and $B=-3x^4y^2$; b) $A=\frac{3}{4}ab^2c$ and $B=\frac{4}{3}a^2bc^3$;

c) $A=-\frac{1}{27}mn^2p^7$, $B=-\frac{3}{2}m^5k^2p$ and $C=-\frac{1}{4}k^4p$.

2. Find the degree and coefficient of the monotone:

a) $(1.5x^2y^3z^4)(4xyz^5)$; b) $³(\frac{7}{8}m^2k^3p^4³)$; c) $(1.5ax^2y^3z^6)(-2.5a^2xy^2z^3)$; d) $\left(-\frac{1}{2}ak^4p^3m^2\right)\left(-\frac{1}{3}a^2bkp^2m^3\right)$.

3. Perform the grading of the ones:

a) $(2x^2y^3z^4)^3$; b) $\left(\frac{2}{3}ab^4c^5\right)^8$; c) $(-bx^4y^5z^2)^{11}$; d) $(-2n^2m^3p^5)^{2n}$.

4. If $u=3abm^2x^3$, $v=-2a^2mxy$ and $w=a^3mxy$, then find $\frac{u.v}{w}.$

5. If $A=(x^2-xy+2y^2)$ and $B=(2x-y)$, then find $A.B$.

6. Prove the identity $(a+b)(c+d)-(a+c)(b+d)-(a-d)(c-b)=0.$

7. Find the numerical value of the expression:

$U=(2mn)^3+3mn^22mn-5mn(mn)^2+2mn^2(-3mn)$ for $m=-\frac{1}{2}, n=\frac{1}{3}$.

8. Find the value of the expression $b(b-1)-b^2+2b$ at $b=-1$.

9. Find the value of the expression $2(3x-2)-x(7-x)$ at $x=-2^2$.

10. Write the expression $(2y-1)(1-y)-(2-y^3)$ with a normal polynomial.

11. Find the value of the expression $A=6(x+5)-2(x-3)(4x-5)+5x(7x-8)-(-6x)^2$ for $x={27^{669}}{(-3)^{2008}}.$

12. Represent the expressions as polynomials in normal form:

(a) $3a+2b-c+(a-3b)-(4a-2c)$; (b) $8-3x-(5-x^2+3x)-(2x+3)$; (c) $(4x^2+2xy+y^2)(2x-y)$.

13. Represent the product $(2x+a)(x^4-5x^3+3x^2-1)$ by a normal polynomial. Find for which value of the parameter $a$ the coefficient of the fourth-degree term and the free term are equal.

14. If $m$ is a parameter and $x$ is a variable, find the value of $m$ for which the polynomial $A=mx^2+3mx^2-2x^3+3x^2-5mx+3m-4$ has a coefficient of the second-degree term equal to $9$.

15. Find the normal form of the expression:

a) $(y-3)(y-1)-(y+1)(y+3)$; b) $3x-2y(x+1)+x(2y-3)$; c) $3m(2m^2+m-1)-2m(m^3+m^2+2)-3$.

16. Let be given the expression $A=b(y^2-2)-(b+3y)(2y-1)$, where $b$ parameter. Represent $A$ by a normal polynomial and find the values of $b$ for which the polynomial:

a) is of degree one; b) has equal coefficients in front of $y^2$ and $y$; c) at $y=1$ has value $0$.

17. Having simplified the expression $M=2(3a-4b)-5[(2a+b)-(a-2b)]-[3(a-b)-6(2a-b)]$, find the value of $M$ if $a=0.1$ and $b=\frac{1}{13}$.

18. Simplify the expression $P=(a^2+2a-1)(a^3-3a^2+a-1)-a(a^4-a^3-6a^2+4a)$ and state at least two values of the variable $a$ for which $P>0$.

19. Find the value of the expression $K=(2-a-a^2)(a^2+3b)+(a^2+b)(a^2-3)+ab(3+2a)$ for $a=-3$ and $b=\frac{1}{3}$.

20. Check the equality: 

a) $x(x-3a)+a(a+x)=9$ for $x=a+3$;

b) $x(x-8a)+a(5x+3)=2$ for $x=3a-1$.

21. Find the normal polynomial identical to the expression $(a^2-3ab-b^2)(5a^2+ab-3b^2)-(a^2-3ab+3b^2)(5a^2+ab+b^2)$.